A four-dimensional body of constant width

TL;DR

The paper constructs a four-dimensional body of constant width by modifying a unit $4$-dimensional Reuleaux simplex $\mathcal{R}$ via a distinguished subset $\mathcal{M}_0$ of its 2-skeleton, yielding $\mathcal{M}=\bigcap_{P\in\mathcal{M}_0}\mathcal{B}(P,1)$ with tetrahedral symmetry. It proves $\mathcal{M}$ has constant width by establishing that $\operatorname{diam}(\mathcal{M}_0)=1$, every boundary point outside $\mathcal{M}_0$ has a unique antipodal point in $\mathcal{M}_0$, and that boundary points are smooth away from $\mathcal{M}_0$, enabling the application of Pal’s theorem. The authors also project $\mathcal{M}$ onto the 3D base hyperplane to obtain a novel tetrahedrally symmetric 3D constant-width body $\pi(\mathcal{M})$ with six elliptical edges and a near-meissner-like volume, estimated at about $0.420$, suggesting it may minimize volume among such 3D bodies. The work deepens the understanding of higher-dimensional constant-width bodies and introduces a new 3D shadow with distinctive non-spherical boundary features, broadening the landscape of extremal convex geometry in higher dimensions.

Abstract

The study of bodies of constant width is a classical subject in convex geometry, with the three-dimensional Meissner bodies being canonical examples. This paper presents a novel geometric construction of a body of constant width in $R^4$, addressing the challenge of constructing such bodies in higher dimensions. Our method produces a natural analogue of the second Meissner body, by modifying a 4-dimensional Reuleaux simplex. The resulting body possesses tetrahedral symmetry and has a boundary composed of both smooth surfaces and a non-smooth subset of the Reuleaux 4-simplex. Furthermore, we analyze the orthogonal projection of this body onto the 3-dimensional hyperplane of its base. This "shadow" is a new 3-dimensional body of constant width with tetrahedral symmetry. It has six elliptical edges and we estimate its volume to be only slightly larger than that of the Meissner bodies.

Figures (9)

• Figure 1: The second Meissner body, the non-smooth circular edges are marked in red.
• Figure 2: The shadow of $\mathcal{M}$.
• Figure 3: Three views of the face $F_{A,B,E}$ being cut by the plane $\mathop{\mathrm{span}}\nolimits(\{A, B\})$ in order to produce $\mathcal{F}_{A,B}$, as seen in the $3$-space generated by $A$, $B$ and $E$.
• Figure 4: The distance from $P$ to $Q$ is less than the distance from $P$ to $Q'$.
• Figure 5: The sets $\mathcal{R}$, $\mathcal{M}$ and $\mathcal{M}_0$ cut by (left) and projected onto (right) the plane $\mathop{\mathrm{span}}\nolimits(\{M_{A,B},M_{C,D}\})$.

Theorems & Definitions (6)

• Theorem 1.1
• Lemma 4.1
• Definition 4.2
• Lemma 4.3
• Proposition 5.1
• proof